Improving force tracking control performance in cooperative robots

Abstract

In order to obtain the environment’s information, cooperative robots could need a lot of sensors. A possible solution to reduce the number of sensors might be the use of control–observer structures. In this article, we have designed a control algorithm by using a modified hybrid computed torque method based on the principle of orthogonalization, but in order to avoid the use of tachometers in the implementation, we are including a velocity observer. The stability proof is developed by using the theory of Lyapunov. Simulation of the proposed control structure compared with a well-known control structure via the performance index analysis is presented. Experimental tests are implemented with the control structure that has the best performance index.

Introduction

Dexterity is one of the most desirable properties when a cooperative robot system is holding and manipulating an object. ^1,2 In order to perform the task, it is necessary to define suitably the desired trajectories and forces to be applied by each end-effector. ³ Robot force control has been studied through decades. Raibert and Craig proposed the so-called hybrid position/force control by introducing a compliance selection matrix which distinguishes position control from force control components in Cartesian coordinates. ^4,5 MaClamroch and Wang gave a proof for local stabilization by using linear feedback for set-point control when both constant target position and contact force are given to the manipulator. ⁶

On the other hand, Arimoto et al. proposed the principle of orthogonalization as an expanded notion of force/position control for robot manipulators under geometric constraints by separating position feedback signals from the force feedback ones by using projection matrices. ⁷ The main idea is to split position/velocity and force errors into two orthogonal spaces, one tangent to the physical constraint at the contact point and the other perpendicular to it. Yun-Hui et al. dealt with decentralized control of multiple manipulators in holonomic cooperation tasks based on the principle of orthogonalization. It should be noticed that a decentralized controller means that each robot is controlled by its own controller. In such a scheme, cooperations between the robots are realized by controlling their interactive forces. ⁸

Among many control techniques that can be found throughout the literature for cooperative systems, Hwang et al. studied grasp force for multi-slave tele-micromanipulation with a single master. ⁹ Similar to Hwang et al., unmanned surface vehicles have also attracted recent attention, that is, adaptive robust finite-time trajectory tracking control of fully actuated marine surface vehicles ¹⁰ ; direct adaptive fuzzy tracking control of marine vehicles with fully unknown parametric dynamics and uncertainties ¹¹ ; adaptive robust online constructive fuzzy control of a complex surface vehicle system ¹² ; and a novel extreme learning control framework of unmanned surface vehicles. ¹³ Murphey et al. considered environment uncertainties, concluding that these effects can be mitigated by using decentralized adaptive techniques. ¹⁴ Moreover, to improve the performance, Rahman and Ikeura took into account that weight perception due to inertia might be different from gravity when lifting an object. ¹⁵ More recently, Rugthum and Tao proposed an adaptive algorithm for cooperative systems in case of actuators’ failures, while guaranteeing asymptotic stability from the closed-loop errors. ¹⁶ In contrast, one of the most important rulings to deal with in position/force control of robots is the possible lack of tachometers and/or force sensors. Many solutions have been proposed in the literature. For instance, Huang and Tseng make use of nonlinear transformations for velocity/force observer design, ¹⁷ while Ohishi et al. applied H^∞ techniques to avoid the use of force sensors. ¹⁸ Observers of the Luenberger type have also been employed, as shown in the literatures. ^19,20 Martínez–Rosas et al. designed a linear velocity observer, avoiding the use of force sensors in an open-loop scheme. ²¹ Later, in 2008, Martínez-Rosas and Arteaga-Pérez improved this approach by including a force observer as well. ²²

In this article, an innovative control law based on the hyperbolic tangent function with a velocity observer is proposed. The algorithm is an improvement over the control approach submitted in the literatures. ^2,5 We compared the operation of both controllers by using the performance index.

This work is organized as follows: The second section presents the dynamic model of the system and their properties. The control structure and the observer are described in the third section. In the fourth section, experimental results are shown. Finally, conclusions are presented in the fifth section.

System model and properties

We consider a cooperative system consisting of l manipulators, each one with nⁱ degrees of freedom and subject to m_i constraints, where n_i > m_i . Suppose the end-effector of the i-manipulator is in touch with an object, as shown in Figure 1. The surface is described by ${\varphi }_i(x_i)=0$ , where x denotes the Cartesian coordinates (task coordinates) fixed at the inertial reference frame. The contact force λ _i arises in the direction of normal vector to the surface through point x and the contact friction arises into the direction of $-\dot{\mathit{x}}$ . Then, the dynamic model of each robot is given by ⁵

Figure 1.

Hybrid control under a geometric constraint.

${\mathit{H}}_i({\mathit{q}}_i){\ddot{\mathit{q}}}_i+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_i+{\mathit{D}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_i+{\mathit{g}}_i({\mathit{q}}_i)={\mathit{\tau }}_i+{\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i){\mathit{\lambda }}_i$

1

where ${\mathit{q}}_i\in {ℝ}^{n_i}$ is the vector of the generalized joint coordinates; ${\mathit{H}}_i({\mathit{q}}_i)\in {ℝ}^{n_i\times n_i}$ is the symmetric positive-definite inertia matrix; ${\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\in {ℝ}^{n_i\times n_i}$ is the matrix of centripetal and Coriolis torques; ${\mathit{g}}_i({\mathit{q}}_i)\in {ℝ}^{n_i}$ is the vector of gravitational torques; ${\mathit{D}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\in {ℝ}^{n_i\times n_i}$ is the positive semi-definite diagonal matrix which represents the viscous friction coefficients; ${\mathit{\tau }}_i\in {ℝ}^{n_i}$ is the vector of input torques acting at the joints; ${\mathit{\lambda }}_i\in {ℝ}^{m_i}$ is the vector of Lagrange multipliers (physically represents the force applied at the contact point), and it is assumed that ${\mathit{J}}_{{\varphi }_i}({\mathit{q}}_i)=\nabla {\mathit{\varphi }}_i({\mathit{q}}_i)\in {ℝ}^{m_i\times n_i}$ is full rank. ^5,7 ∇ ϕ _i ( q _i ) denotes the gradient of the surface’s object ${\mathit{\varphi }}_i({\mathit{q}}_i)\in {ℝ}^{m_i}$ satisfying

${\mathit{\varphi }}_i({\mathit{q}}_i)=0$

2

and which maps a vector onto the normal plane to the tangent plane that arises at the contact point. ^2,5 This relationship means that homogeneous constraints are being considered. ^2,5

Assuming for simplicity that all robots have got revolute joints, we can establish the following properties:

Property 1. H _i ( q _i ) satisfies ${\mathit{\lambda }}_{{\text{h}}_i}\parallel {\mathit{x}}_i{\parallel }^2\le {\mathit{x}}_i^{\text{T}}{\mathit{H}}_i({\mathit{q}}_i){\mathit{x}}_i\le {\mathit{\lambda }}_{{\text{H}}_i}\parallel {\mathit{x}}_i{\parallel }^2\hspace{0.17em}\forall {\mathit{q}}_i$ , ${\mathit{x}}_i\in {ℝ}^{n_i}$ where ${\mathit{\lambda }}_{{\text{h}}_i}\stackrel{\Delta }{{\displaystyle =}}\underset{\forall {\mathit{q}}_i\in {ℝ}^{n_i}}{\text{min}}{\mathit{\lambda }}_{{\text{min}}_i}({\mathit{H}}_i({\mathit{q}}_i))$ , ${\mathit{\lambda }}_{{\text{H}}_i}\stackrel{\Delta }{{\displaystyle =}}\underset{\forall {\mathit{q}}_i\in {ℝ}^{n_i}}{{\displaystyle \text{max}}}{\mathit{\lambda }}_{{\text{max}}_i}({\mathit{H}}_i({\mathit{q}}_i))$ , and $0<{\mathit{\lambda }}_{{\text{h}}_i}\le {\mathit{\lambda }}_{{\text{H}}_i}<\infty$ . ^2,5 ▲

Property 2. With a proper definition of ${\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)$ , the matrix $\left[{\dot{\mathit{H}}}_i({\mathit{q}}_i)-2{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\right]$ is skew-symmetric. ^2,5 ▲

Property 3. The vector C _i ( q _i , x _i ) y _i satisfies ${\mathit{C}}_i({\mathit{q}}_i,{\mathit{x}}_i){\mathit{y}}_i={\mathit{C}}_i({\mathit{q}}_i,{\mathit{y}}_i){\mathit{x}}_i\hspace{0.17em}\forall {\mathit{x}}_i,{\mathit{y}}_i\in {ℝ}^{n_i}$ . ^2,5 ▲

Property 4. The vector ${\dot{\mathit{q}}}_i$ can be rewritten as

${\dot{\mathit{q}}}_i={\mathit{Q}}_i({\mathit{q}}_i){\dot{\mathit{q}}}_i+{\mathit{P}}_i({\mathit{q}}_i){\dot{\mathit{q}}}_i$

3

where ${\mathit{P}}_i({\mathit{q}}_i)={\mathit{J}}_{{\varphi }_i}^{+}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}({\mathit{q}}_i)\in {ℝ}^{n_i\times n_i}$ and ${\mathit{Q}}_i({\mathit{q}}_i)=\mathit{I}-{\mathit{P}}_i({\mathit{q}}_i)\in {ℝ}^{n_i\times n_i}$ with rank ( q _i ) = n_i − m_i ; I is the identity matrix; ${\mathit{J}}_{{\varphi }_i}^{+}({\mathit{q}}_i)={\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i){\left[{\mathit{J}}_{{\varphi }_i}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)\right]}^{-1}\in {ℝ}^{n_i\times m_i}$ is the Moore–Penrose pseudo-inverse. ${\mathit{J}}_{{\varphi }_i}^{+}({\mathit{q}}_i)$ and Q _i ( q _i ) are orthogonal, that is, ${\mathit{Q}}_i({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{+}({\mathit{q}}_i)=0$ and ${\mathit{Q}}_i({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)=0$ . Note that since ϕ _i ( q _i ) = 0, it holds ${\mathit{J}}_{{\varphi }_i}({\mathit{q}}_i){\dot{\mathit{q}}}_i=0$ and ${\mathit{Q}}_i({\mathit{q}}_i){\dot{\mathit{q}}}_i={\dot{\mathit{q}}}_i$ . ^2,5 ▲

Remark 1

In workspace coordinates, constraint (2) becomes

${\mathit{\varphi }}_{\mathit{\text{x}}i}(x_i)=0$

4

and equivalently ${\mathit{J}}_{\varphi \text{x}i}({\mathit{q}}_i)=\nabla {\mathit{\varphi }}_{\text{x}i}({\mathit{x}}_i)\in {ℝ}^{m_i\times n_i}$ . Therefore, we have

${\dot{\mathit{x}}}_i={\mathit{Q}}_{\text{x}i}({\mathit{x}}_i){\dot{\mathit{x}}}_i+{\mathit{P}}_{\text{x}i}({\mathit{x}}_i){\dot{\mathit{x}}}_i={\mathit{Q}}_{\text{x}i}({\mathit{x}}_i){\dot{\mathit{x}}}_i$

5

with ${\mathit{Q}}_{\text{x}i}({\mathit{x}}_i)=\mathit{I}-{\mathit{J}}_{\varphi \text{x}i}^{+}({\mathit{q}}_i){\mathit{J}}_{\varphi \text{x}i}({\mathit{q}}_i)$ where rank $({\mathit{Q}}_{\text{x}i})=n_i-m_i$ ; ${\mathit{P}}_{\text{x}i}={\mathit{J}}_{\varphi \text{x}i}^{+}({\mathit{x}}_i){\mathit{J}}_{\varphi \text{x}i}({\mathit{x}}_i)$ . Once again ${\mathit{J}}_{\varphi \text{x}i}^{+}({\mathit{x}}_i)$ and Q _xi ( x _i ) are orthogonal. Supposing now that a desired trajectory fulfills constraint (4), that is, ϕ _xi ( x _di ) = 0. For this case, the tracking error can be defined as

$\begin{array}{c}{\tilde{\mathit{x}}}_i={\mathit{x}}_i-{\mathit{x}}_{\text{d}i}\\ ={\mathit{Q}}_{\text{x}i}({\mathit{x}}_i){\mathit{x}}_i+{\mathit{P}}_{\text{x}i}({\mathit{x}}_i){\mathit{x}}_i-{\mathit{Q}}_{\text{x}i}({\mathit{x}}_{\text{d}i}){\mathit{x}}_{\text{d}i}-{\mathit{P}}_{\text{x}i}({\mathit{x}}_{\text{d}i}){\mathit{x}}_{\text{d}i}\end{array}$

6

In Figure 2(a), the case for a large tracking error is shown, while in Figure 2(b), it can be seen that ${\tilde{\mathit{x}}}_i$ tends to be tangent to the surface as it decreases. ⁶ This means that for small enough errors, there must be possible to obtain the following approximation

Figure 2.

Plane tangent to the surface (tracking error). (a) Large tracking error and (b) small tracking error.

${\tilde{\mathit{x}}}_i\approx {\mathit{Q}}_{\text{x}i}({\mathit{x}}_i)\left[{\mathit{x}}_i-{\mathit{x}}_{\text{d}i}\right]={\mathit{Q}}_{\text{x}i}({\mathit{x}}_i){\tilde{\mathit{x}}}_i$

7

where ${\mathit{Q}}_{\text{x}i}({\mathit{x}}_i)\in {ℝ}^{n_i\times n_i}$ projected ${\tilde{\mathit{x}}}_i\in {ℝ}^{n_i}$ into the position’s subspace. Furthermore, from equation (5) it should also hold

${\dot{\tilde{\mathit{x}}}}_i\approx {\mathit{Q}}_{\text{x}i}({\mathit{x}}_i)\left[{\dot{\mathit{x}}}_i-{\dot{\mathit{x}}}_{\text{d}i}\right]={\mathit{Q}}_{\text{x}i}({\mathit{x}}_i){\dot{\tilde{\mathit{x}}}}_i$

8

This means that a local stability analysis can be performed in a small enough region around ${\tilde{\mathit{x}}}_i=0$ , where equations (7) and (8) hold. Note that for simplicity’s sake workspace coordinates are used in this remark to illustrate this fact, but just we can get to the very same conclusion for joint coordinates, that is,

${\tilde{\mathit{q}}}_i\approx {\mathit{Q}}_i({\mathit{q}}_i){\tilde{\mathit{q}}}_i$

9

${\dot{\tilde{\mathit{q}}}}_i\approx {\mathit{Q}}_i({\mathit{q}}_i){\dot{\tilde{\mathit{q}}}}_i$

10

hold for a small enough error ${\tilde{\mathit{q}}}_i$ . $▲$

In order to design the control-observer scheme, the following assumption is established. ²¹

Assumption 1

The contact between the object and each end-effector is firm, and it is made at one single point, as shown in Figure 1. This means that the relative movement between the surfaces does not exist and that the robots of the cooperative system satisfy the constraint ϕ _i ( q _i ) = 0 through all time.▲

It should be noticed that assumption 1 is made for simplicity, but in practice, the control law must guarantee a firm contact with the object just by pushing it.

Control and observer design

In order to design the control law, based on the literatures ^2,5 we consider the following auxiliary variables:

${\tilde{\mathit{\lambda }}}_i\stackrel{\Delta }{=}{\mathit{\lambda }}_i-{\mathit{\lambda }}_{\text{d}i}$

11

${\tilde{\mathit{F}}}_i\stackrel{\Delta }{=}{\mathit{F}}_i-{\mathit{F}}_{\text{d}i}$

12

${\mathit{F}}_i\stackrel{\Delta }{=}{\displaystyle \underset{0}{\overset{t}{\int }}}{\mathit{\lambda }}_i(\mathit{\tau })\text{d}\mathit{\tau }$

13

${\mathit{F}}_{\text{d}i}(t)\stackrel{\Delta }{=}{\displaystyle \underset{0}{\overset{t}{\int }}}{\mathit{\lambda }}_{\text{d}i}(\mathit{\tau })\text{d}\mathit{\tau }$

14

${\tilde{\mathit{q}}}_i\stackrel{\Delta }{=}{\mathit{q}}_i-{\mathit{q}}_{\text{d}i}$

15

${\dot{\mathit{q}}}_{{\text{o}}_i}\stackrel{\Delta }{=}{\dot{\widehat{\mathit{q}}}}_i-{\mathbf{\Lambda }}_i{\mathit{z}}_i$

16

${\mathit{r}}_i\stackrel{\Delta }{=}{\dot{\mathit{q}}}_i-{\dot{\mathit{q}}}_{{\text{o}}_i}={\dot{\mathit{z}}}_i+{\Lambda }_i{\mathit{z}}_i$

17

${\mathit{s}}_i\stackrel{\Delta }{=}{\dot{\mathit{q}}}_i-{\dot{\mathit{q}}}_{{\text{r}}_i}$

18

${\mathit{z}}_i\stackrel{\Delta }{=}{\mathit{q}}_i-{\widehat{\mathit{q}}}_i$

19

where ${\mathit{q}}_{\text{d}i}\in {ℝ}^{n_i}$ is a desired smooth bounded trajectory satisfying ${\mathit{\varphi }}_i({\mathit{q}}_{\text{d}i})=0$ , $(\widehat{\cdot })$ represents the estimated value of (⋅), and ${\mathit{\lambda }}_{\text{d}i}\in {ℝ}^{m_i}$ is the desired force to be applied by each manipulator on the object. ²²

The nominal reference signal is defined as

${\dot{\mathit{q}}}_{{\text{r}}_i}\stackrel{\Delta }{=}{\mathit{Q}}_i({\mathit{q}}_i)\left[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i})\right]+{\mathit{J}}_{\varphi i}^{+}({\mathit{q}}_i){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)$

20

where ${\mathit{\xi }}_i>\mathit{O}\in {ℝ}^{m_i\times m_i}$ and ${\mathbf{\Lambda }}_i>\mathit{O}\in {ℝ}^{n_i\times n_i}$ are diagonal matrices, ${\mathit{J}}_{{\varphi }_i}^{+}({\mathit{q}}_i)={\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i){\left[{\mathit{J}}_{{\varphi }_i}({\mathit{q}}_i)J_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)\right]}^{-1}\in {ℝ}^{n_i\times m_i}$ is the Moore–Penrose pseudo inverse and $\text{tanh}({\tilde{\mathit{F}}}_i)={\left[\text{tanh}({\tilde{F}}_{i1}),\dots ,\text{tanh}({\tilde{F}}_{i{m}_i})\right]}^{\text{T}}$ , with ${\tilde{F}}_{ij}$ element of ${\tilde{\mathit{F}}}_i$ for $j=1,\dots ,m_i$ . By manipulating equatiuon (20), using equation (3) and substituting in equation (18), we have

$\begin{array}{c}{\mathit{s}}_i={\mathit{Q}}_i({\mathit{q}}_i){\dot{\mathit{q}}}_i-{\mathit{Q}}_i({\mathit{q}}_i)[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i})]-{\mathit{J}}_{\varphi i}^{+}({\mathit{q}}_i){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\\ =\underset{{\mathit{s}}_{\text{p}i}}{\underbrace{{\mathit{Q}}_i({\mathit{q}}_i)[{\dot{\tilde{\mathit{q}}}}_i+{\Lambda }_i{\tilde{\mathit{q}}}_i-{\mathbf{\Lambda }}_i{\mathit{z}}_i]}}+\underset{{\mathit{s}}_{{\text{f}}_i}}{\underbrace{(-J_{{\varphi }_i}^{+}({\mathit{q}}_i){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i))}}\end{array}$

21

where ${\mathit{s}}_{{\text{p}}_i}$ is orthogonal to ${\mathit{s}}_{{\text{f}}_i}$ .

By computing the derivative of equation (20), we obtain

$\begin{array}{c}{\ddot{\mathit{q}}}_{{\text{r}}_i}={\dot{\mathit{Q}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\left[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i}\right)\right]+{\dot{\mathit{J}}}_{\varphi i}^{+}\left({\mathit{q}}_i,{\dot{\mathit{q}}}_i\right){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\\ +{\mathit{Q}}_i({\mathit{q}}_i)\left[{\ddot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\dot{\widehat{\mathit{q}}}}_i-{\dot{\mathit{q}}}_{\text{d}i}\right)\right]+{\mathit{J}}_{\varphi i}^{+}({\mathit{q}}_i){\mathit{\xi }}_i{\text{sech}}^2({\tilde{\mathit{F}}}_i){\tilde{\mathit{\lambda }}}_i\end{array}$

22

where ${\text{sech}}^2({\tilde{\mathit{F}}}_i)$ is a diagonal matrix whose elements are given by ${\text{sech}}^2({\tilde{F}}_{ij})$ for $j=1,\dots ,m_i$ , and ${\dot{\mathit{Q}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\in {ℝ}^{n_i\times n_i}$ is defined as follows:

${\dot{Q}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)=\left[\begin{array}{ccc}\frac{\mathit{\partial }{a}_{11}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_i& \dots & \frac{\mathit{\partial }{a}_{1n_i}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_i\\ ⋮& \ddots & ⋮\\ \frac{\mathit{\partial }{a}_{n_{i}1}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_i& \dots & \frac{\mathit{\partial }{a}_{n_i{n}_i}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_i\end{array}\right]$

23

with a_jk elements of Q _i ( q _i ),j,k = 1,…,n_i , as in the literatures. ^2,5

Based on equation (23), we define

${\dot{\widehat{\mathit{Q}}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{o}}_i})=\left[\begin{array}{ccc}\frac{\mathit{\partial }{a}_{11}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_{{\text{o}}_i}& \dots & \frac{\mathit{\partial }{a}_{1n_i}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_{{\text{o}}_i}\\ ⋮& \ddots & ⋮\\ \frac{\mathit{\partial }{a}_{n_{i}1}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_{{\text{o}}_i}& \dots & \frac{\mathit{\partial }{a}_{n_i{n}_i}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{\dot{\mathit{q}}}_{{\text{o}}_i}\end{array}\right]$

24

where ${\dot{\mathit{q}}}_{{\text{o}}_i}\in {ℝ}^{n_i}$ is given in equation (16), as in the literatures. ^2,5 By considering equations (23) and (24), the following error matrix can be calculated:

$\begin{array}{c}{\dot{\stackrel{\sim }{\mathit{Q}}}}_i(r_i)={\dot{\mathit{Q}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)-{\dot{\widehat{\mathit{Q}}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{o}}_i})\\ ={\dot{\stackrel{\sim }{\mathit{Q}}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i-{\dot{\mathit{q}}}_{{\text{o}}_i})\\ =\left[\begin{array}{ccc}\frac{\mathit{\partial }{a}_{11}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{r}_i& \dots & \frac{\mathit{\partial }{a}_{1n_i}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{r}_i\\ ⋮& \ddots & ⋮\\ \frac{\mathit{\partial }{a}_{n_{i}1}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{r}_i& \dots & \frac{\mathit{\partial }{a}_{n_i{n}_i}({\mathit{q}}_i)}{\mathit{\partial }{\mathit{q}}_i}{r}_i\end{array}\right]\end{array}$

25

where ${\mathit{r}}_i\in {ℝ}^{n_i}$ is defined in equation (17), as in the literatures. ^2,5 By considering equations (22) and (24), we can define ${\ddot{\widehat{\mathit{q}}}}_{{\text{r}}_i}\in {ℝ}^{n_i}$ as follows:

$\begin{array}{c}{\ddot{\widehat{\mathit{q}}}}_{{\text{r}}_i}\underset{ }{\overset{\Delta }{=}}{\dot{\widehat{\mathit{Q}}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{o}}_i})\left[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i}\right)\right]+{\dot{\widehat{\mathit{J}}}}_{\varphi i}^{+}({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{o}}_i}){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\\ +{\mathit{Q}}_i({\mathit{q}}_i)\left[{\ddot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\dot{\widehat{\mathit{q}}}}_i-{\dot{\mathit{q}}}_{\text{d}i}\right)\right]+{\mathit{J}}_{\varphi i}^{+}({\mathit{q}}_i){\mathit{\xi }}_i{\text{sech}}^2({\tilde{\mathit{F}}}_i){\tilde{\mathit{\lambda }}}_i\end{array}$

26

where ${\dot{\widehat{\mathit{J}}}}_{{\varphi }_i}^{+}\left({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{o}}_i}\right)\in {ℝ}^{n_i\times m_i}$ is defined likewise $\dot{\widehat{\mathit{Q}}}\left({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{o}}_i}\right)$ in equation (24). By manipulating equation (26), we obtain

$\begin{array}{c}{\ddot{\widehat{\mathit{q}}}}_{{\text{r}}_i}={\dot{\mathit{Q}}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\left[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i}\right)\right]+{\dot{\mathit{J}}}_{\varphi i}^{+}({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)+{\mathit{Q}}_i({\mathit{q}}_i)\left[{\ddot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\dot{\widehat{\mathit{q}}}}_i-{\dot{\mathit{q}}}_{\text{d}i}\right)\right]+{\mathit{J}}_{\varphi i}^{+}({\mathit{q}}_i){\mathit{\xi }}_i{\text{sech}}^2({\tilde{\mathit{F}}}_i){\tilde{\mathit{\lambda }}}_i\\ -{\dot{\stackrel{\sim }{\mathit{Q}}}}_i(r_i)\left[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i}\right)\right]-{\dot{\stackrel{\sim }{\mathit{J}}}}_{\varphi i}^{+}(r_i){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\\ ={\ddot{\mathit{q}}}_{{\text{r}}_i}+\mathit{e}\left({\mathit{r}}_i\right)\end{array}$

27

where ${\dot{\stackrel{\sim }{\mathit{J}}}}_{\varphi i}^{+}({\mathit{r}}_i)\in {ℝ}^{n_i\times m_i}$ is defined in the same way as ${\dot{\stackrel{\sim }{\mathit{Q}}}}_i({\mathit{r}}_i)$ in equation (25), and $\mathit{e}\left({\mathit{r}}_i\right)\in {ℝ}^{n_i}$ is defined as follows:

$\mathit{e}({\mathit{r}}_i)\stackrel{\Delta }{=}-{\dot{\stackrel{\sim }{\mathit{Q}}}}_i({\mathit{r}}_i)\left[{\dot{\mathit{q}}}_{\text{d}i}-{\mathbf{\Lambda }}_i\left({\widehat{\mathit{q}}}_i-{\mathit{q}}_{\text{d}i}\right)\right]-{\dot{\stackrel{\sim }{\mathit{J}}}}_{\varphi i}^{+}({\mathit{r}}_i){\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)$

28

Proposed controller

The control law is given by

${\mathit{\tau }}_i={\mathit{H}}_i({\mathit{q}}_i){\ddot{\widehat{\mathit{q}}}}_{{\text{r}}_i}+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\dot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{D}}_i{\dot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{g}}_i({\mathit{q}}_i)-{\mathit{K}}_{{\text{R}}_i}\left({\dot{\mathit{q}}}_{\text{o}i}-{\dot{\mathit{q}}}_{{\text{r}}_i}\right)-{\mathit{J}}_{\varphi i}^{\text{T}}({\mathit{q}}_i)\left[{\mathit{\lambda }}_{\text{d}i}-{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\right]$

29

where ${\mathit{K}}_{{\text{R}}_i}\in {ℝ}^{n_i\times n_i}$ is a positive-definite diagonal matrix. ^2,5 Substituting equation (28) in equation (29), we obtain

${\mathit{\tau }}_i={\mathit{H}}_i({\mathit{q}}_i){\ddot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{H}}_i({\mathit{q}}_i){\mathit{e}}_i({\mathit{r}}_i)+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\dot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{D}}_i{\dot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{g}}_i({\mathit{q}}_i)-{\mathit{J}}_{\varphi i}^{\text{T}}({\mathit{q}}_i)\left[{\mathit{\lambda }}_{\text{d}i}-{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\right]-{\mathit{K}}_{{\text{R}}_i}\left({\dot{\mathit{q}}}_{\text{o}i}-{\dot{\mathit{q}}}_{{\text{r}}_i}\right)$

30

By applying the proposed control given in equation (30) into the dynamic model described in equation (1), considering the fact that ${\mathit{s}}_i-{\mathit{r}}_i={\dot{\mathit{q}}}_{\text{o}i}-{\dot{\mathit{q}}}_{{\text{r}}_i}$ , and substituting equation (18), we obtain

${\mathit{H}}_i({\mathit{q}}_i){\dot{\mathit{s}}}_i+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_i-{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\dot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{D}}_i{\mathit{s}}_i+{\mathit{K}}_{{\text{R}}_i}{\mathit{s}}_i={\mathit{J}}_{\varphi i}^{\text{T}}({\mathit{q}}_i)[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)]+{\mathit{K}}_{{\text{R}}_i}{\mathit{r}}_i+{\mathit{H}}_i({\mathit{q}}_i){\mathit{e}}_i({\mathit{r}}_i)$

31

Taking into account property 3, we have

$\begin{array}{c}{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_i-{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\dot{\mathit{q}}}_{{\text{r}}_i}={\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_i-{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_{{\text{r}}_i}+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\dot{\mathit{q}}}_{{\text{r}}_i}-{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\dot{\mathit{q}}}_{{\text{r}}_i}\\ ={\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)\left[{\dot{\mathit{q}}}_i-{\dot{\mathit{q}}}_{{\text{r}}_i}\right]+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i})\left[{\dot{\mathit{q}}}_i-{\dot{\mathit{q}}}_{{\text{r}}_i}\right]\\ ={\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\mathit{s}}_i+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\mathit{s}}_i\end{array}$

32

By substituting equation (32) in equation (31), we get the closed-loop error dynamics as

${\mathit{H}}_i({\mathit{q}}_i){\dot{\mathit{s}}}_i={\mathit{K}}_{{\text{R}}_i}{\mathit{r}}_i+{\mathit{J}}_{\varphi i}^{\text{T}}({\mathit{q}}_i)[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)]-{\mathit{K}}_{{\text{DR}}_i}{\mathit{s}}_i-{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i}){\mathit{s}}_i-{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i){\mathit{s}}_i+{\mathit{H}}_i({\mathit{q}}_i){\mathit{e}}_i({\mathit{r}}_i)$

33

where

${\mathit{K}}_{{\text{DR}}_i}\stackrel{\Delta }{=}{\mathit{D}}_i+{\mathit{K}}_{{\text{R}}_i}$

34

From (33), we obtain

$\begin{array}{c}{\dot{\mathit{s}}}_i-{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{J}}_{\varphi i}^{\text{T}}({\mathit{q}}_i)\left[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\right]-{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{K}}_{{\text{R}}_i}{\mathit{r}}_i-{\mathit{e}}_i({\mathit{r}}_i)\\ =-{\mathit{H}}_i^{-1}({\mathit{q}}_i)\left[{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i})+{\mathit{C}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i)+{\mathit{K}}_{{\text{DR}}_i}\right]{\mathit{s}}_i\\ \underset{ }{\overset{\Delta }{=}}{\mathit{f}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i},{\mathit{s}}_i)\end{array}$

35

Observer definition

We propose the following observer:

${\dot{\widehat{\mathit{q}}}}_i={\dot{\widehat{\mathit{q}}}}_{{\text{o}}_i}+{\mathbf{\Lambda }}_i{\mathit{z}}_i+{\mathit{K}}_{\text{d}i}{\mathit{z}}_i$

36

${\ddot{\widehat{\mathit{q}}}}_{{\text{o}}_i}={\ddot{\widehat{\mathit{q}}}}_{{\text{r}}_i}+{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)\left[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\right]+{\mathit{K}}_{\text{d}i}{\mathbf{\Lambda }}_i{\mathit{z}}_i$

37

where ${\dot{\widehat{\mathit{q}}}}_{{\text{o}}_i}\in {ℝ}^{n_i}$ and ${\mathbf{\Lambda }}_i,{\mathit{K}}_{\text{d}i}\in {ℝ}^{n_i\times n_i}$ are positive-definite matrices. ⁵ By computing the derivative of equation (36) and substituting the result in equation (37), we obtain

${\ddot{\widehat{\mathit{q}}}}_i-{\mathbf{\Lambda }}_i{\dot{\mathit{z}}}_i-{\mathit{K}}_{\text{d}i}{\dot{\mathit{z}}}_i={\ddot{\widehat{\mathit{q}}}}_{{\text{r}}_i}+{\mathit{K}}_{\text{d}i}{\mathbf{\Lambda }}_i{\mathit{z}}_i+{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)\left[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)\right]$

38

By substituting equation (27) into equation (38), we obtain

${\ddot{\mathit{q}}}_{{\text{r}}_i}={\ddot{\widehat{\mathit{q}}}}_i-{\mathbf{\Lambda }}_i{\dot{\mathit{z}}}_i-{\mathit{K}}_{\text{d}i}({\dot{\mathit{z}}}_i+{\mathbf{\Lambda }}_i{\mathit{z}}_i)-\mathit{e}({\mathit{r}}_i)-{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)]$

39

or

${\ddot{\mathit{q}}}_i-{\ddot{\mathit{q}}}_{{\text{r}}_i}={\ddot{\mathit{q}}}_i-{\ddot{\widehat{\mathit{q}}}}_i+{\mathbf{\Lambda }}_i{\dot{\mathit{z}}}_i+{\mathit{K}}_{\text{d}i}({\dot{\mathit{z}}}_i+{\mathbf{\Lambda }}_i{\mathit{z}}_i)+\mathit{e}({\mathit{r}}_i)+{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)]$

40

By considering equations (16), (17), and (18), we can rewrite equation (40) after equation (35) as follows:

$\begin{array}{c}{\dot{\mathit{r}}}_i+{\mathit{K}}_{\text{d}i}{\mathit{r}}_i={\dot{\mathit{s}}}_i-{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{J}}_{{\varphi }_i}^{\text{T}}({\mathit{q}}_i)[{\tilde{\mathit{\lambda }}}_i+{\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)]-\mathit{e}({\mathit{r}}_i)\\ ={\mathit{f}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i},{\mathit{s}}_i)+{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{K}}_{{\text{R}}_i}{\mathit{r}}_i\end{array}$

41

Finally, we obtain the following closed-loop dynamics for the observer error

${\dot{\mathit{r}}}_i+{\mathit{K}}_{\text{dh}i}{\mathit{r}}_i={\mathit{f}}_i({\mathit{q}}_i,{\dot{\mathit{q}}}_i,{\dot{\mathit{q}}}_{{\text{r}}_i},{\mathit{s}}_i)$

42

with

${\mathit{K}}_{\text{dh}i}\stackrel{\Delta }{=}{\mathit{K}}_{\text{d}i}-{\mathit{H}}_i^{-1}({\mathit{q}}_i){\mathit{K}}_{{\text{R}}_i}$

43

For the dynamic error system described by equations (12), (33), and (42), the state is defined as follows:

${\mathit{x}}_i\stackrel{\Delta }{{\displaystyle =}}\left[\begin{array}{c}{\mathit{s}}_i\\ {\mathit{r}}_i\\ {\tilde{\mathit{F}}}_i\end{array}\right]$

44

Consider now the following lemma:

Lemma 1

Assume that ${\mathit{Q}}_i({\mathit{q}}_i){\mathbf{\Lambda }}_i={\mathbf{\Lambda }}_i{\mathit{Q}}_i({\mathit{q}}_i)$ and define a region

${\mathcal{D}}_i=\{{\mathit{x}}_i:\parallel xi\parallel \le {\mathit{x}}_{{\text{max}}_i}\}$

45

where ${\mathit{x}}_{{\text{max}}_i}$ is small enough for relations (9) and (10) to hold, then ${\tilde{\mathit{q}}}_i,{\dot{\tilde{\mathit{q}}}}_i,{\mathit{z}}_i,{\dot{\mathit{z}}}_i$ and ${\tilde{\mathit{F}}}_i$ are bounded too. Furthermore, if ${\mathit{x}}_i$ tends to zero, then ${\tilde{\mathit{q}}}_i,{\dot{\tilde{\mathit{q}}}}_i,{\mathit{z}}_i,{\dot{\mathit{z}}}_i$ and ${\tilde{\mathit{F}}}_i$ tend to zero as well.

Proof: It is rather obvious that r _i and ${\tilde{\mathit{F}}}_i$ are bounded and tend to zero according to x _i . Also, after equation (17), ${\mathit{z}}_i,{\dot{\mathit{z}}}_i$ must be bounded and tend to zero as well. As to ${\tilde{\mathit{q}}}_i,{\dot{\tilde{\mathit{q}}}}_i$ consider equation (21), that is, ${\mathit{s}}_i={\mathit{s}}_{{\text{p}}_i}+{\mathit{s}}_{{\text{f}}_i}$ . Then, ${\mathit{s}}_{{\text{p}}_i}$ must be bounded since it is orthogonal to ${\mathit{s}}_{{\text{f}}_i}$ , which is bounded. Thus, ${\mathit{s}}_{{\text{p}}_i}$ tends to zero if x _i does, which lies in ${\mathcal{D}}_i$ . This means that the following approximation holds

${\mathit{s}}_{{\text{p}}_i}={\mathit{Q}}_i({\mathit{q}}_i)[{\dot{\tilde{\mathit{q}}}}_i+{\mathbf{\Lambda }}_i{\tilde{\mathit{q}}}_i-{\mathbf{\Lambda }}_i{\mathit{z}}_i]\approx {\dot{\tilde{\mathit{q}}}}_i+{\mathbf{\Lambda }}_i{\tilde{\mathit{q}}}_i-{\mathbf{\Lambda }}_i{\mathit{Q}}_i({\mathit{q}}_i){\mathit{z}}_i$

46

so that we can conclude that both ${\tilde{\mathit{q}}}_i,{\dot{\tilde{\mathit{q}}}}_i$ are bounded and tend to zero whenever s _i does. ▲

Note that for any x i in the ball (45), s _i , r _i , and ${\tilde{\mathit{F}}}_i$ are bounded. In view of definition (17), we obtain that both z _i and ${\dot{\mathit{z}}}_i$ are bounded.

Remark 2

The region ${\mathcal{D}}_i$ is defined as shown in equation (45) for simplicity’s sake, since the norm of ${\tilde{\mathit{F}}}_i$ is also required to be small, while this is not necessary at all in order for equations (9) and (10) to hold. Furthermore, notice that it is trivial to obtain ${\mathit{Q}}_i({\mathit{q}}_i){\mathbf{\Lambda }}_i={\mathbf{\Lambda }}_i{\mathit{Q}}_i({\mathit{q}}_i)$ just by setting Λ _i as a positive constant. Finally, how small $x_{{\text{max}}_i}$ must depend on the surface, but it DOES NOT necessarily means $x_{{\text{max}}_i}\approx 0$ . ▲

Theorem 1

Consider the closed-loop dynamics (12), (33), and (42) formed by the cooperative system described in equation (1), the control law in equation (29), and the observer defined in equations (36) and (37), for i = 1,…,l. Assume that ${\mathit{Q}}_i({\mathit{q}}_i){\mathbf{\Lambda }}_i={\mathbf{\Lambda }}_i{\mathit{Q}}_i({\mathit{q}}_i)$ . Then, control-observer gains can always be found to guarantee that ${\tilde{\mathit{q}}}_i,{\dot{\tilde{\mathit{q}}}}_i,{\mathit{z}}_i,{\dot{\mathit{z}}}_i,{\tilde{\mathit{F}}}_i$ and ${\tilde{\mathit{\lambda }}}_i$ remain bounded and tend to zero as long as the initial error conditions are chosen small enough to remain in $l$ regions satisfying the conditions of Lemma 1. ▲

The proof of theorem 1 is given in the Appendix 1, where a proper region of attraction is found.

Remark 3

The result of theorem 1 is only local within a possible (very) small region ${\mathcal{D}}_i$ . This is to ensure the convergence of the tracking errors, ${\tilde{\mathit{q}}}_i$ and ${\dot{\tilde{\mathit{q}}}}_i$ , to zero. However, it is not a strong drawback because before holding the object the robots are in free movement, and once it is hold, all initial errors can be set to zero just by choosing ${\mathit{q}}_{{\text{d}}_i}(0)={\mathit{q}}_i(0)$ and assuming desired, estimated, and actual velocities to be zero at t = 0. It is worth pointing out that the controller-observer scheme is implemented for each robot separately, while only the knowledgement of every constraint ϕ_i ( q _i ) = 0 is required. ^2,5 ▲

Simulation and experimental results

In order to test the theory presented in the previous section, we have done some simulations and some experimental tests. The simulations were performed to compare the control structure proposed in the literatures, ^2,5 and the control structure presented throughout this article. The experiments were carried out with the control structure that had a better performance according to the performance index analysis.

System description

The cooperative system consists of two industrial robots from CRS robotics (Figure 3). Robots are the CRS-A255 (5 DOF) and the CRS-A465 (6 DOF). It should be noticed that only the first three joints of each robot are used, while the remaining joints are mechanically braked. Each joint is driven by a DC motor with optical encoders, whose dynamics have been taken into account for control-observer implementation as explained by Gudiño Lau and Arteaga-Pérez. ²³ The system’s framework is at A465 robot’s base. Both manipulators get a crash protection device on the end-effector, and a force sensor installed on it. Each end-effector is an interchangeably multi-tool, and it is set on the sensor. The experiments were performed on a personal computer with a Pentium IV 1.4 GHz processor with two PCI-FlexMotion-6C boards from National Instruments. The sampling time is 9 ms. The controllers are programmed in LabWindows/CVI software from National Instruments. The object is a cube of white melamine about 0.400 kg. The cube’s dimensions are 0.15 × 0.15 × 0.311 m³.

Figure 3.

Cooperative system.

Simulation results

The task is to pick up an object with a desired force and move it into a plane. The corresponding constraints given in Cartesian coordinates are simply described by

${\varphi }_i(x_i)=x_i-d_i(t)=0\hspace{1em}\text{for}\hspace{1em}i=1,2$

47

where $x_i\in ℝ$ is the scalar $x$ coordinate of the $i$ -robot, and $d_i(t)\in {ℝ}_{+}$ is a positive constant with a difference given by the object width.

It should be noticed that any desired trajectory should be chosen in order to fulfill the system’s constraints, and in this way, it is also easy enough to get a zero initial error whenever the initial velocities are null. For the simulation, it is then proposed

$x_{d1}=0.5530\hspace{0.17em}(\text{m})$

48

$x_{d2}=0.8522\hspace{0.17em}(\text{m})$

49

$y_{d1,2}=0.0095\text{sin}\left(\mathit{\omega }\left(t-t_i\right)\right)\hspace{0.17em}(\text{m})$

50

$z_{d1,2}=\left[0.635+0.0095\text{cos}\left(\mathit{\omega }\left(t-t_i\right)\right)-0.0095\right]\hspace{0.17em}(\text{m})$

51

where ω(t) is a fifth-order polynomial designed to satisfy ω(t _i) = 0 and ω(t _f) = 2π for an initial time t_i and a final time t _f. In this way, the robots will make a circle in the yz plane. Inverse kinematics of the manipulators are used to compute q _di , where i = 1, 2. As it was explained before, the proposed control-observer scheme is an improvement over the algorithm presented by Gudiño et al. ^2,5 and in the simulations it is used for comparison purposes. Notice that we use the same gain notation. Since β_i in the literatures ^2,5 does not have any equivalent in this article, we have set it to zero without affecting control performance. The complete set of gains is shown in Table 1.

Table 1.

Control and observer parameters.

Parameter	Value	Parameter	Value
${\mathbf{\Lambda }}_1$	$21.034\hspace{0.17em}\mathit{I}$	${\mathbf{\Lambda }}_2$	$24.270\hspace{0.17em}I$
${\mathit{K}}_{R_1}$	$\text{diag}\hspace{0.17em}\text{{161}.\text{8}\hspace{0.17em}161.8\hspace{0.17em}\text{161}.8\}$	${\mathit{K}}_{R_2}$	$\text{diag}\hspace{0.17em}\text{{64}.72\hspace{0.17em}\text{48}.54\hspace{0.17em}\text{32}.36\}$
${\mathit{K}}_{d1}$	6.472 I	${\mathit{K}}_{d2}$	6.472 I

On the other hand, the main modification on the control law is the inclusion of the function ${\mathit{\xi }}_i\text{tanh}({\tilde{\mathit{F}}}_i)$ in the force error term, which acts as a saturated signal for large errors. Since we have m_i = 1, ξ _i is actually a scalar. First, we choose for the proposed algorithm and for what has been given in the study by Gudiño et al., ^2,5 the value ξ = 1 for i = 1, 2, and show that is as simple as the modification that may have been, it is enough to clearly improve force tracking performance. Then, we carry out a simulation just for the proposed algorithm with a higher value, ξ _i = 5, and show that the performance is even better because a faster convergence to zero of the error signal is achieved. Note that much higher values may decrease the transient performance. It can be seen in Figure 4(a) and (b) that the position errors for both manipulators are basically the same for any value of ξ _i .

Figure 4.

Position errors in Cartesian coordinates $(x_2-x_{\text{d}2},y_2-y_{\text{d}2},z_2-z_{\text{d}2})$ . ${\xi }_i=1$ , ( $---$ ) for the proposed algorithm and $(-\cdot -\cdot -)$ for Gudiño et al.’ study.^2,5 ${\xi }_i=5$ , (——–) for the proposed algorithm. (a) Robot A255 and (b) robot A465.

On the other hand, in Figure 5(a) and (b), the effect of the new force control term on the system performance can be appreciated. The corresponding force errors are shown in Figure 6(a) and (b), where the improvement can be seen more easily.

Figure 5.

Force tracking. ${\xi }_i=1$ , ( $---$ ) for the proposed algorithm and $(-\cdot -\cdot -)$ for Gudiño et al.’s study.^2,5 ${\xi }_i=5$ , (——–) for the proposed algorithm. Desired force ( $\cdot \hspace{0.17em}\cdot \hspace{0.17em}\cdot \hspace{0.17em}\cdot \hspace{0.17em}\cdot$ ). (a) Robot A255 and (b) robot A465.

Figure 6.

Force error. ${\xi }_i=1$ , ( $---$ ) for the proposed algorithm and $(-\cdot -\cdot -)$ for Gudiño et al.’s study.^2,5 ${\xi }_i=5$ , (——–) for the proposed algorithm. (a) Robot A255 and (b) robot A465.

Performance index

Robot manipulators are complex mechanical systems due to nonlinear and multivariable nature of their dynamic behavior. For this reason, in the robotics community, there are no well-established criteria for proper evaluation of the controllers. However, from a practical point of view, comparing the performance of the controllers using standard scalar value of ${\mathcal{L}}^2$ norm, which is an objective measure of a numerical error curve, it is accepted. ${\mathcal{L}}^2$ norm measures the average root-mean square for the error and is given by:

${\mathcal{L}}^2\left[\left(\tilde{\cdot }\right)\right]=\sqrt{\frac{1}{t-t_0}{\displaystyle \underset{t_0}{\overset{t}{\int }}}{\Vert \left(\tilde{\cdot }\right)\Vert }^2\text{d}t}$

52

where $\left(\tilde{\cdot }\right)\in {ℝ}^{n_i}$ represents an error function of the analyzed physical parameter (position, force, etc.), and $t,t_0\in {ℝ}_{+}$ are the initial and final times, respectively. ²⁴ A small value of ${\mathcal{L}}^2$ represents a small error of the variable analyzed; therefore, the smallest value of ${\mathcal{L}}^2$ represents the smallest error indicating the best performance. The overall results are summarized in Figure 7, which include the performance indexes of the controllers analyzed. ²⁴ For a better understanding, the data presented in Figure 7 are compared with respect to the standard controller ${\mathcal{L}}^2$ presented in the literatures. ^2,5 As shown in Figure 7, it can be concluded that the control structure with better performance is the proposed control structure, equation (29), with ξ _i = 5.

Figure 7.

Performance index. (a) Position ${\mathcal{L}}^2\hspace{0.17em}({\tilde{q}}_i)$ and (b) Force ${\mathcal{L}}^2\hspace{0.17em}({\tilde{F}}_i)$ .

Experimental results

We conducted experiments by using the control structure with the best performance index. In this case, the proposed control structure with ξ _i = 5 was used. The experiment consists of carrying robots from the home position to the contact point at the object. They hold the object with a desired force and lift it to follow a circular path. We consider in this experiment that the location and weight of the object are known. Table 2 shows the known values we use in the experiment.

Table 2.

Experimental parameters.

Parameter	Value
Desired applied force	$25+15\text{sin}([2\pi t]/tf)$ (N)
Desired trajectory	Circular trajectory of radio = 0.05 (m)
Initial point	$\left(0.4928,0.5019\right)$ (m)
Run time	16.5 (s)

We can see in Figure 8 that the cooperative system follows the desired trajectory successfully. Furthermore, in Figure 9, we can see that both robots apply the right desired force on the object.

Figure 8.

Trajectories.

Figure 9.

Forces (applied, desired, and error).

The position errors in Cartesian coordinate are shown in Figures 10 and 11.

Figure 10.

Position errors in Cartesian coordinate (robot A255).

Figure 11.

Position errors in Cartesian coordinate (robot A465).

Since we proposed a velocity observer, and this was used in the design of control, then its behavior is presented in Figures 12 and 13.

Figure 12.

Velocity observer (robot A255).

Figure 13.

Velocity observer (robot A465).

Conclusions

The force and position tracking control without velocity measurements in cooperative systems formed by two robot manipulators holding an object is addressed in this work. The control law is a decentralized approach, which considers constrained movement directions in order to omit to know the dynamics of the object. It is assumed that the robot’s dynamics are known and that contact forces are measurable using a force sensor at the end-effector. In order to reduce the number of sensors used in a cooperative system, no velocity measurements are made, instead a velocity observer is proposed in order to obtain the information. The proposed scheme is based on the structure defined in the literatures, ^2,5 with consideration to improve the force term. We focus on the term of force because the main objective is to hold a body firmly without breaking and follow a trajectory, for this reason the proposed control structure enhances the force term by using the hyperbolic tangent function. This modification improves performance in the force applied to hold the object and track the desired trajectory without dropping the body. By simulations, it is shown that force performance can be improved without decreasing position tracking. The results have been compared with the original algorithm, and the performance improvement is clear. Experimental results show the robustness of the proposed approach whenever robot model parameters are uncertain.

It is worth mentioning that the proposed solution is based on the properties of the hyperbolic tangent function, which is a real function, saturated, bijective in the condominium, odd, and strictly increasing.

Future work

Future work is to estimate the mass of the manipulated object and automatically adjust the force applied to avoid damaging the object.